Properties

Label 2-3200-8.5-c1-0-58
Degree $2$
Conductor $3200$
Sign $i$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.317i·3-s + 2.89·9-s − 3.78i·11-s + 1.89·17-s − 5.97i·19-s − 1.87i·27-s − 1.20·33-s − 6.79·41-s + 8.48i·43-s − 7·49-s − 0.603i·51-s − 1.89·57-s − 14.1i·59-s − 16.3i·67-s + 15.6·73-s + ⋯
L(s)  = 1  − 0.183i·3-s + 0.966·9-s − 1.14i·11-s + 0.460·17-s − 1.37i·19-s − 0.360i·27-s − 0.209·33-s − 1.06·41-s + 1.29i·43-s − 49-s − 0.0845i·51-s − 0.251·57-s − 1.84i·59-s − 1.99i·67-s + 1.83·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $i$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.760272664\)
\(L(\frac12)\) \(\approx\) \(1.760272664\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 0.317iT - 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 3.78iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 1.89T + 17T^{2} \)
19 \( 1 + 5.97iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 6.79T + 41T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 16.3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 17.0iT - 83T^{2} \)
89 \( 1 - 4.10T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.287198514017754229458927179117, −7.890042634629577817163274401356, −6.80216703826085228549045146696, −6.46409983543521407964348469810, −5.34781597212091731735130827024, −4.69243630231824038262369112038, −3.66421006401019417141165307053, −2.89140266603604311305672398330, −1.67845939906725181624099824744, −0.57128140795724239532338152535, 1.32375590742799428679056681339, 2.15792020858860269943023734412, 3.47641566163354016037920353399, 4.16670901801288976727145591440, 4.95762288531116673030986609288, 5.76718477696392616676563607441, 6.75486194952162976879698813938, 7.34004554813528478615838108404, 8.015990568919252171893311842259, 8.889786572883767456920230322597

Graph of the $Z$-function along the critical line